integral of (3x^2-7x)^5(6x-7)

解答

\int (3 x^{2} - 7 x)^{5} (6 x - 7) d x

\frac{243 x ^{12}}{2} - 1701 x^{11} + \frac{19845 x ^{10}}{2} - 30870 x^{9} + \frac{108045 x ^{8}}{2} - 50421 x^{7} + \frac{117649 x ^{6}}{6} + C

求解步骤

\int (3 x^{2} - 7 x)^{5} (6 x - 7) d x

乘开

(3 x^{2} - 7 x)^{5} (6 x - 7) : 1458 x^{11} - 18711 x^{10} + 99225 x^{9} - 277830 x^{8} + 432180 x^{7} - 352947 x^{6} + 117649 x^{5}

= \int 1458 x^{11} - 18711 x^{10} + 99225 x^{9} - 277830 x^{8} + 432180 x^{7} - 352947 x^{6} + 117649 x^{5} d x

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 1458 x^{11} d x - \int 18711 x^{10} d x + \int 99225 x^{9} d x - \int 277830 x^{8} d x + \int 432180 x^{7} d x - \int 352947 x^{6} d x + \int 117649 x^{5} d x

\int 1458 x^{11} d x = \frac{243 x ^{12}}{2}

\int 18711 x^{10} d x = 1701 x^{11}

\int 99225 x^{9} d x = \frac{19845 x ^{10}}{2}

\int 277830 x^{8} d x = 30870 x^{9}

\int 432180 x^{7} d x = \frac{108045 x ^{8}}{2}

\int 352947 x^{6} d x = 50421 x^{7}

\int 117649 x^{5} d x = \frac{117649 x ^{6}}{6}

= \frac{243 x ^{12}}{2} - 1701 x^{11} + \frac{19845 x ^{10}}{2} - 30870 x^{9} + \frac{108045 x ^{8}}{2} - 50421 x^{7} + \frac{117649 x ^{6}}{6}

解答补常数

= \frac{243 x ^{12}}{2} - 1701 x^{11} + \frac{19845 x ^{10}}{2} - 30870 x^{9} + \frac{108045 x ^{8}}{2} - 50421 x^{7} + \frac{117649 x ^{6}}{6} + C

\int x (e^{(3 x - 2)} + 5) d x

ma c l a u r in tan (θ)

x \to 9 + lim (\frac{∣ x - 9 ∣}{x - 9})

x \to 0 lim (cot (\frac{π}{2} - x))

\int 4 sin (x) cos^{2} (x) d x